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--- en:grundlagenhandbuch:massetemperatur [2026/05/24 17:07] – [Dimensionless characteristics for the temperature estimation] neelest
+++ en:grundlagenhandbuch:massetemperatur [2026/06/05 20:18] (aktuell) – [Determination of the Average Temperature] deppe2
@@ Zeile 60: / Zeile 60: @@
 ==== Solution for the temperature estimation ====
-$$\Theta(\zeta, \xi) = \left[\Theta_0 + \frac{Br}{Gz} \zeta + \frac{Br}{Gz} \xi^2\right] \text{erf}\left(\frac{\xi}{2\sqrt{\frac{\zeta}{Gz}}}\right) + Br \xi \sqrt{\frac{\zeta}{Gz}} e^{[-\frac{Gz\xi^2}{4\zeta}]} - \frac{Br}{2} \xi^2$$
+$$\Theta(\zeta, \xi) = \left[\Theta_0 + \frac{Br}{Gz} \zeta + \frac{Br}{Gz} \xi^2\right] \text{erf}\left(\frac{\xi}{2\sqrt{\frac{\zeta}{Gz}}}\right) + Br \xi \sqrt{\frac{\zeta}{Gz}} e^{[-\frac{Gz\xi^2}{4\zeta}]} - \frac{Br}{2} \xi^2\tag{9}$$
-$$\bar{\Theta} = \frac{\bar{T} - T_Z}{T_Z} = \int_0^1 \Theta(\zeta, \xi) d\xi$$
+$$\bar{\Theta} = \frac{\bar{T} - T_Z}{T_Z} = \int_0^1 \Theta(\zeta, \xi) d\xi\tag{10}$$
 In every heating zone a constant wall temperature is imposed. The temperature calculation starts at the point where the first melt is found. This is the place of the first melt pool formation, at the start of the melting respectively.
@@ Zeile 68: / Zeile 68: @@
 The average start temperature $\bar{\Theta}_{Start}$ is calculated using the energy equation for the melt layer on the barrel wall. The energy equation for the melt layer on the barrel wall is [[en:grundlagenhandbuch:massetemperatur#references |[Pot91]]], [[en:grundlagenhandbuch:massetemperatur#references |[Sch90]]]:
-$$\lambda_s \frac{\partial^2 T}{\partial y^2} + \tau_{yi} \frac{\partial v_j}{\partial y} = 0 \tag{6}$$
+$$\lambda_s \frac{\partial^2 T}{\partial y^2} + \tau_{yi} \frac{\partial v_j}{\partial y} = 0 \tag{11}$$
 With the boundary conditions $T(0) = T_{Fl}$ und $T(\delta) = T_Z$ one reaches the following solution:
-$$T(\xi) = T_{Fl} + (T_Z - T_{Fl}) \left\{\xi + Br \left[\frac{1}{A^2}\left(\frac{A}{e^A - 1}\right)^{1+n}\left(1 - e^{A\xi} - \xi(1 - e^A)\right)\right]\right\} \tag{7}$$
+$$T(\xi) = T_{Fl} + (T_Z - T_{Fl}) \left\{\xi + Br \left[\frac{1}{A^2}\left(\frac{A}{e^A - 1}\right)^{1+n}\left(1 - e^{A\xi} - \xi(1 - e^A)\right)\right]\right\} \tag{12}$$
 with:
-$$Br = \frac{K(T_{Fl})v_{rel}^{1+n} \overline{\delta}^{1-n}}{\lambda_s (T_Z - T_{Fl})} \tag{8}$$
+$$Br = \frac{K(T_{Fl})v_{rel}^{1+n} \overline{\delta}^{1-n}}{\lambda_s (T_Z - T_{Fl})} \tag{13}$$
 and
-$$\xi = \frac{y}{\delta} \tag{9}$$
+$$\xi = \frac{y}{\delta} \tag{14}$$
-$$A = \frac{\beta(T_Z - T_{Fl})}{n} \tag{10}$$
+$$A = \frac{\beta(T_Z - T_{Fl})}{n} \tag{15}$$
 From it we get the average temperature for the melt layer under the consideration of the velocity profile:
-$$\bar{T} = T_{Start} = \frac{\int_0^1 v(\xi)T(\xi)d\xi}{\int_0^1 v(\xi)d\xi} \tag{11}$$
+$$\bar{T} = T_{Start} = \frac{\int_0^1 v(\xi)T(\xi)d\xi}{\int_0^1 v(\xi)d\xi} \tag{16}$$
 and
-$$v(\xi) = v_{rel} \frac{e^{A\xi} - 1}{e^A - 1} \tag{12}$$
+$$v(\xi) = v_{rel} \frac{e^{A\xi} - 1}{e^A - 1} \tag{17}$$
 ==== Equations to estimate the average starting temperature ====
-$$T_{Start} = [B \Delta T(A - 1)2e^{2A} - (AB \Delta T e^{2A} + A_3 - A2)]A_4$$
+$$T_{Start} = [B \Delta T(A - 1)2e^{2A} - (AB \Delta T e^{2A} + A_3 - A2)]A_4\tag{18}$$
-$$B = Br \frac{1}{A^2} \left[\frac{A}{e^A - 1}\right]^{1+n} \qquad \Delta T = T_Z - T_{Fl} A = \frac{\beta}{n} \Delta T$$
+$$B = Br \frac{1}{A^2} \left[\frac{A}{e^A - 1}\right]^{1+n} \qquad \Delta T = T_Z - T_{Fl} A = \frac{\beta}{n} \Delta T\tag{19}$$
-$$A_1 = A^2(B + 1) + 3AB + 2(B - 1)$$
+$$A_1 = A^2(B + 1) + 3AB + 2(B - 1)\tag{20}$$
-$$A_2 = [\Delta T(A(B + 1) + 2B - 1) + AT_{Fl}]2e^A$$
+$$A_2 = [\Delta T(A(B + 1) + 2B - 1) + AT_{Fl}]2e^A\tag{21}$$
-$$A_3 = A^2B \Delta T e^A + \Delta T A_1 + 2AT_{Fl}(A + 1)$$
+$$A_3 = A^2B \Delta T e^A + \Delta T A_1 + 2AT_{Fl}(A + 1)\tag{22}$$
-$$A_4 = \frac{1}{2A(e^A - A - 1)}$$
+$$A_4 = \frac{1}{2A(e^A - A - 1)}\tag{23}$$
 As Solution one obtains the equations, which are listed in the table. The average temperature for the melt layer is the same as that of the average temperature for the temperature flow estimation. Under the assumption $Br = 0$ the average starting temperature can be solved with the simple solution [[en:grundlagenhandbuch:massetemperatur#references |[Pot91]]], [[en:grundlagenhandbuch:massetemperatur#references |[TK78]]]:
-$$\bar{T}_{\mathrm{start}} = T_{\mathrm{Fl}} + (T_Z - T_{\mathrm{Fl}}) \cdot \frac{\frac{1}{A} - 1 + e^A \left(1 - \frac{1}{A}\right)}{e^A - A - 1}$$
+$$\bar{T}_{\mathrm{start}} = T_{\mathrm{Fl}} + (T_Z - T_{\mathrm{Fl}}) \cdot \frac{\frac{1}{A} - 1 + e^A \left(1 - \frac{1}{A}\right)}{e^A - A - 1}\tag{24}$$
 ===== Modified Model =====
@@ Zeile 124: / Zeile 124: @@
 $$\rho \cdot c \left( \frac{\partial T}{\partial t} + v_x \frac{\partial T}{\partial x} + v_y \frac{\partial T}{\partial y} + v_z \frac{\partial T}{\partial z} \right) = \left( \frac{\partial \dot{q}_x}{\partial x} + \frac{\partial \dot{q}_y}{\partial y} + \frac{\partial \dot{q}_z}{\partial z} \right) $$
 $$- T \left( \frac{\partial p}{\partial T} \right)_{\rho} \left( \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} \right) - \left[ \tau_{xx} \frac{\partial v_x}{\partial x} + \tau_{yy} \frac{\partial v_y}{\partial x} + \tau_{zz} \frac{\partial v_z}{\partial x} \right] $$
-$$- \left[ \tau_{xy} \left( \frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x} \right) + \tau_{zx} \left( \frac{\partial v_x}{\partial z} + \frac{\partial v_z}{\partial x} \right) + \tau_{zy} \left( \frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y} \right) \right] \tag{1}$$
+$$- \left[ \tau_{xy} \left( \frac{\partial v_x}{\partial y} + \frac{\partial v_y}{\partial x} \right) + \tau_{zx} \left( \frac{\partial v_x}{\partial z} + \frac{\partial v_z}{\partial x} \right) + \tau_{zy} \left( \frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y} \right) \right] \tag{25}$$
@@ Zeile 141: / Zeile 141: @@
 Under these assumptions the energy equation simplifies to:
-$$\frac{\partial T}{\partial z} = -\frac{1}{\rho \cdot c \cdot \bar{v}_{0z}} \cdot \frac{\partial \dot{q}_y}{\partial y} + \frac{\overline{ ({\tau} \cdot {\gamma})_j }}{\rho \cdot c \cdot \bar{v}_{0z}} \cdot e^{-\beta (T - T_j)} \tag{2}$$
+$$\frac{\partial T}{\partial z} = -\frac{1}{\rho \cdot c \cdot \bar{v}_{0z}} \cdot \frac{\partial \dot{q}_y}{\partial y} + \frac{\overline{ ({\tau} \cdot {\gamma})_j }}{\rho \cdot c \cdot \bar{v}_{0z}} \cdot e^{-\beta (T - T_j)} \tag{26}$$
 For the heat flow the Fourier's law for heat conduction may apply
-$$\dot{q}_y = -\lambda \frac{\partial T}{\partial y} \tag{3}$$
+$$\dot{q}_y = -\lambda \frac{\partial T}{\partial y} \tag{27}$$
 Is the differential equation standardized, then follows in conjunction with the dimensionless parameters defined in the table:
-$$\frac{\partial \Theta}{\partial \zeta} = \frac{1}{Gz} \cdot \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} e^{-\beta T_z \Theta} \tag{4}$$
+$$\frac{\partial \Theta}{\partial \zeta} = \frac{1}{Gz} \cdot \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} e^{-\beta T_z \Theta} \tag{28}$$
 **Table:** Dimensionless factors for the temperature estimation.
@@ Zeile 161: / Zeile 161: @@
 | $ Gz $ | $ \frac{ c \cdot \rho \cdot \bar{h} \cdot \dot{V}_{z} }{ \lambda \bar{b} \Delta z } $ |
-Due to the exponential term the partial differential equation (4) cannot be solved analytically. In accordance to Melisch [[en:grundlagenhandbuch:massetemperatur#references |[Mel98]]] this term is linearized. The following equation results:
+Due to the exponential term the partial differential equation 28 cannot be solved analytically. In accordance to Melisch [[en:grundlagenhandbuch:massetemperatur#references |[Mel98]]] this term is linearized. The following equation results:
-$$\frac{\partial \Theta}{\partial \zeta} = \frac{1}{Gz} \cdot \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} \cdot (C_1 - C_2 \cdot \beta \cdot T_z \cdot \Theta) \tag{5}$$
+$$\frac{\partial \Theta}{\partial \zeta} = \frac{1}{Gz} \cdot \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} \cdot (C_1 - C_2 \cdot \beta \cdot T_z \cdot \Theta) \tag{29}$$
 The factors $C_1$ and $C_2$ are defined in [[en:grundlagenhandbuch:massetemperatur#references |[Mel98]]]. The solution of this differential equation describes the temperature change for a zone with constant geometry as well as invariant material properties (except the influence of the viscosity). To comply with these requirements the estimation has to be performed in small sections.
@@ Zeile 169: / Zeile 169: @@
 ==== The Channel Area ====
-For the resolution of the equation (1) the starting temperature $T_0$ of the mathematical interval is selected as reference temperature. The coordinate system is laid on the screw root surface.
+For the resolution of the equation 30 the starting temperature $T_0$ of the mathematical interval is selected as reference temperature. The coordinate system is laid on the screw root surface.
 For the resolution of the differential equation it is assumed that the temperature differences in an interval $\Delta z$ are diminutive. The temperature equalization processes, which are to be expected in direction of the screw channel depth, are therefore approximately constant and almost independent of the z-coordinate for a certain calculation section.
-$$\frac{\partial^2 \Theta}{\partial \xi^2} \approx \frac{\partial^2 \Theta_0}{\partial \xi^2} \tag{1}$$
+$$\frac{\partial^2 \Theta}{\partial \xi^2} \approx \frac{\partial^2 \Theta_0}{\partial \xi^2} \tag{30}$$
-Consequently, the alteration of the heat flow in $\xi$-direction can be estimated in a simplifying way on the basis of the starting temperature profile. A potential equation for the specification of the temperature profile in direction of the screw channel depth is Equation (2).
+Consequently, the alteration of the heat flow in $\xi$-direction can be estimated in a simplifying way on the basis of the starting temperature profile. A potential equation for the specification of the temperature profile in direction of the screw channel depth is Equation 31.
-$$\Theta_0 = \Theta_z + (K_1 + K_2) \cdot \xi^2 + K_1 \cdot \xi^4 + K_2 \cdot \xi^6 \tag{2}$$
+$$\Theta_0 = \Theta_z + (K_1 + K_2) \cdot \xi^2 + K_1 \cdot \xi^4 + K_2 \cdot \xi^6 \tag{31}$$
 The following boundary conditions were applied for the resolution of the differential equation:
-  * Heat transfer to the screw does not take place: $\frac{\partial \Theta}{\partial \xi} = 0$ für $\xi = 0 \tag{3}$
+  * Heat transfer to the screw does not take place: $\frac{\partial \Theta}{\partial \xi} = 0$ für $\xi = 0$
   * A predetermined temperature $T_{z}$ is found at the barrel wall:
-$$\Theta(\zeta, \xi = 1) = \Theta_z = \frac{T_z - T_0}{T_z} \tag{4}$$
+$$\Theta(\zeta, \xi = 1) = \Theta_z = \frac{T_z - T_0}{T_z} \tag{32}$$
   * The medium melt temperature is known at the beginning of the calculation section:
-$$\int_0^1 \Theta(\zeta = 0, \xi) \cdot d \xi = \frac{T_0 - T_0}{T_z} = 0 \tag{5}$$
+$$\int_0^1 \Theta(\zeta = 0, \xi) \cdot d \xi = \frac{T_0 - T_0}{T_z} = 0 \tag{33}$$
-From this the following solution of equation (1) derives:
+From this the following solution of equation 30 derives:
 $$\Theta(\zeta, \xi) = \frac{2 \cdot K_1 - 2 \cdot K_2 + 12 \cdot K_1 \cdot \xi^2 + 30 \cdot K_2 \cdot \xi^4 + Br \cdot C_1}{Br \cdot C_2 \cdot \beta \cdot T_z} \cdot (1 - \varepsilon)$$
-$$+ (\Theta_z + (K_1 - K_2) \cdot \xi^2 +) \cdot K_1 \cdot \xi^4 + K_2 \cdot \xi^6) \cdot \varepsilon \tag{6}$$
+$$+ (\Theta_z + (K_1 - K_2) \cdot \xi^2 +) \cdot K_1 \cdot \xi^4 + K_2 \cdot \xi^6) \cdot \varepsilon \tag{34}$$
 with
-$$K_1 = -\frac{5}{2} \cdot \frac{147 \cdot (\Theta_z - \varepsilon) + Br \cdot C_2 \cdot (1 - \varepsilon) + Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z \cdot (\varepsilon - 1)}{231 \cdot (1 - \varepsilon) + 5 \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z} \tag{7}$$
+$$K_1 = -\frac{5}{2} \cdot \frac{147 \cdot (\Theta_z - \varepsilon) + Br \cdot C_2 \cdot (1 - \varepsilon) + Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z \cdot (\varepsilon - 1)}{231 \cdot (1 - \varepsilon) + 5 \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z} \tag{35}$$
-$$K_2 = \frac{7}{4} \cdot \frac{105 \cdot \Theta_z \cdot (1 - \varepsilon) + 4 \cdot Br \cdot C_2 \cdot (\varepsilon - 1) + Br \cdot Gz \cdot \beta \cdot T_z \cdot \Theta_z \cdot (4 + 11 \cdot \varepsilon)}{231 \cdot (1 - \varepsilon) + 5 \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z} \tag{8}$$
+$$K_2 = \frac{7}{4} \cdot \frac{105 \cdot \Theta_z \cdot (1 - \varepsilon) + 4 \cdot Br \cdot C_2 \cdot (\varepsilon - 1) + Br \cdot Gz \cdot \beta \cdot T_z \cdot \Theta_z \cdot (4 + 11 \cdot \varepsilon)}{231 \cdot (1 - \varepsilon) + 5 \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z} \tag{36}$$
 The medium temperature in the channel is calculated with:
-$$\varepsilon = e^{\left(-\frac{Br \cdot C_2 \cdot \beta \cdot T_z}{Gz} \zeta\right)} \tag{9}$$
+$$\varepsilon = e^{\left(-\frac{Br \cdot C_2 \cdot \beta \cdot T_z}{Gz} \zeta\right)} \tag{37}$$
-$$\overline{\Theta} = \frac{\overline{T} - T_0}{T_z} = \int_0^1 \Theta(\xi, \zeta) \cdot d \xi \tag{10}$$
+$$\overline{\Theta} = \frac{\overline{T} - T_0}{T_z} = \int_0^1 \Theta(\xi, \zeta) \cdot d \xi \tag{38}$$
 This results in:
@@ Zeile 213: / Zeile 213: @@
 $$\overline{\Theta} = \frac{K_2 \cdot \varepsilon}{7} + \frac{C_1 \cdot (1 - \varepsilon)}{C_2 \cdot \beta \cdot T_z} + \Theta_z \cdot \varepsilon + K_1 \cdot \frac{6 - 6 \cdot \varepsilon + \frac{8}{15} \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z}{Br \cdot C_2 \cdot \beta \cdot T_z}$$
-$$+K_2 \cdot \frac{4 - 4 \cdot \varepsilon - \frac{1}{3} \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z}{Br \cdot C_2 \cdot \beta \cdot T_z} \tag{11}$$
+$$+K_2 \cdot \frac{4 - 4 \cdot \varepsilon - \frac{1}{3} \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z}{Br \cdot C_2 \cdot \beta \cdot T_z} \tag{39}$$
 For a first verification the model predictions were confronted with the results of non-isothermal flow simulations. The figure exhibits the comparison of the medium temperatures. An improvement of the description quality can be observed here as compared to the models published so far.
@@ Zeile 231: / Zeile 231: @@
 The temperature increase in the radial clearance and in the grooves can be calculated with the same approach as the one, with which the temperature increase is determined in the channel.
-$$\frac{\partial T}{\partial x} = -\frac{1}{\rho \cdot c \cdot \overline{v}_{0x}} \cdot \frac{\partial \dot q}{\partial y} + \frac{\overline{(\tau \cdot \dot{\gamma})}_j}{\rho \cdot c \cdot \overline{v}_{0x}} \cdot e^{-\beta(T-T_i)} \tag{1}$$
+$$\frac{\partial T}{\partial x} = -\frac{1}{\rho \cdot c \cdot \overline{v}_{0x}} \cdot \frac{\partial \dot q}{\partial y} + \frac{\overline{(\tau \cdot \dot{\gamma})}_j}{\rho \cdot c \cdot \overline{v}_{0x}} \cdot e^{-\beta(T-T_i)} \tag{40}$$
 Is the differential equation standardized, then follows in conjunction with the dimensionless parameters defined in the table:
-$$\frac{\partial \Theta}{\partial \zeta} = \frac{1}{Gz} \cdot \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} e^{-\beta \cdot T_z \cdot \Theta} \tag{2}$$
+$$\frac{\partial \Theta}{\partial \zeta} = \frac{1}{Gz} \cdot \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} e^{-\beta \cdot T_z \cdot \Theta} \tag{41}$$
 The resolution of the differential equation is performed analogous to chapter The Channel Area. One receives:
@@ Zeile 241: / Zeile 241: @@
 $$\overline{\Theta} = \frac{K_2 \cdot \varepsilon}{7} + \frac{C_1 \cdot (1 - \varepsilon)}{C_2 \cdot \beta \cdot T_z} + \Theta_z \cdot \varepsilon + K_1 \cdot \frac{6 - 6 \cdot \varepsilon + \frac{8}{15} \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z}{Br \cdot C_2 \cdot \beta \cdot T_z}$$
-$$+K_2 \cdot \frac{4 - 4 \cdot \varepsilon - \frac{1}{3} \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z}{Br \cdot C_2 \cdot \beta \cdot T_z} \tag{3}$$
+$$+K_2 \cdot \frac{4 - 4 \cdot \varepsilon - \frac{1}{3} \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z}{Br \cdot C_2 \cdot \beta \cdot T_z} \tag{42}$$
 with:
-$$K_1 = -\frac{5}{2} \cdot \frac{147 \cdot (\Theta_z - \varepsilon) + Br \cdot C_2 \cdot (1 - \varepsilon) + Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z \cdot (\varepsilon - 1)}{231 \cdot (1 - \varepsilon) + 5 \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z} \tag{4}$$
+$$K_1 = -\frac{5}{2} \cdot \frac{147 \cdot (\Theta_z - \varepsilon) + Br \cdot C_2 \cdot (1 - \varepsilon) + Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z \cdot (\varepsilon - 1)}{231 \cdot (1 - \varepsilon) + 5 \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z} \tag{43}$$
-$$K_2 = \frac{7}{4} \cdot \frac{105 \cdot \Theta_z \cdot (1 - \varepsilon) + 4 \cdot Br \cdot C_2 \cdot (\varepsilon - 1) + Br \cdot Gz \cdot \beta \cdot T_z \cdot \Theta_z \cdot (4 + 11 \cdot \varepsilon)}{231 \cdot (1 - \varepsilon) + 5 \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z} \tag{5}$$
+$$K_2 = \frac{7}{4} \cdot \frac{105 \cdot \Theta_z \cdot (1 - \varepsilon) + 4 \cdot Br \cdot C_2 \cdot (\varepsilon - 1) + Br \cdot Gz \cdot \beta \cdot T_z \cdot \Theta_z \cdot (4 + 11 \cdot \varepsilon)}{231 \cdot (1 - \varepsilon) + 5 \cdot \varepsilon \cdot Br \cdot C_2 \cdot \beta \cdot T_z \cdot \Theta_z} \tag{44}$$
 and:
-$$\varepsilon = e^{\left(\frac{Br \cdot C_2 \cdot \beta \cdot T_z}{Gz} \cdot \zeta\right)} \tag{6}$$
+$$\varepsilon = e^{\left(\frac{Br \cdot C_2 \cdot \beta \cdot T_z}{Gz} \cdot \zeta\right)} \tag{45}$$
 **Dimensionless factors for the temperature calculation in the radial clearance and in the grooves:**
@@ Zeile 267: / Zeile 267: @@
 The coupling of the temperature calculations in the channel area and the clearance area [[en:grundlagenhandbuch:massetemperatur#references |[Ste92]]] can be performed by means of a balance of the enthalpy being fed and the enthalpy streaming off.
-{{ :grundlagenhandbuch:massetemperatur:modifizierter_ansatz:de_sigma150_dlg_grundlagenhandbuch_massetemperatur_004.png?nolink |}}
+{{ :en:grundlagenhandbuch:de_sigma150_dlg_grundlagenhandbuch_massetemperatur_006.svg?nolink&600 |}}
 **Figure:** Control room for the enthalpy balance to calculate the medium temperature.
@@ Zeile 273: / Zeile 273: @@
 The enthalpy change in the control room corresponds with the difference between the inflowing and the outflowing enthalpy:
-$$\Delta \dot{H}_{z_0,1} = \dot{H}_{z_1} - \dot{H}_{z_0} \tag{1}$$
+$$\Delta \dot{H}_{z_0,1} = \dot{H}_{z_1} - \dot{H}_{z_0} \tag{46}$$
 The alteration results from the temperature increase in the channel plus the enthalpy change of the flows in the radial clearance and in the grooves.
-$$\Delta \dot{H}_{z_0,1} = \rho \cdot c \cdot \dot{V}_z \cdot (T_{z1,channel} - T_{z,0}) + \Delta \dot{H}_{gap} + \Delta \dot{H}_{groove} \tag{2}$$
+$$\Delta \dot{H}_{z_0,1} = \rho \cdot c \cdot \dot{V}_z \cdot (T_{z1,channel} - T_{z,0}) + \Delta \dot{H}_{gap} + \Delta \dot{H}_{groove} \tag{47}$$
 with:
-$$\Delta \dot{H}_{zgap} = \rho \cdot c \cdot \dot{V}_x \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{gap}) \tag{3}$$
+$$\Delta \dot{H}_{zgap} = \rho \cdot c \cdot \dot{V}_x \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{gap}) \tag{48}$$
-$$\Delta \dot{H}_{zgroove} = \rho \cdot c \cdot \dot{V}_{groove} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{4}$$
+$$\Delta \dot{H}_{zgroove} = \rho \cdot c \cdot \dot{V}_{groove} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{49}$$
 $\overline{T_{z01,i}}$ is the integrally averaged temperature in the control room:
-$$\overline{T_{z01,i}} = \frac{1}{\Delta z} \cdot \int_{z_0}^{z_1} T_{channel}(z) \cdot dz \tag{5}$$
+$$\overline{T_{z01,i}} = \frac{1}{\Delta z} \cdot \int_{z_0}^{z_1} T_{channel}(z) \cdot dz \tag{50}$$
 Is it simplifyingly assumed that the isotherms run at right angle to the screw axis, the medium temperature in the neighboring channel $\overline{T_{z01,i+1}}$ can be determined:
-$$\overline{T_{z01,i+1}} = \frac{1}{\Delta z} \cdot \int_{z_0+\frac{t}{\sin(\varphi_s)}}^{z_1+\frac{t}{\sin(\varphi_s)}} T_{channel}(z) \cdot dz \tag{6}$$
+$$\overline{T_{z01,i+1}} = \frac{1}{\Delta z} \cdot \int_{z_0+\frac{t}{\sin(\varphi_s)}}^{z_1+\frac{t}{\sin(\varphi_s)}} T_{channel}(z) \cdot dz \tag{51}$$
 The overall temperature increase in the interval $[z_0, z_1]$ thus calculates to:
-$$T_{z_1} = \frac{\dot{H}_{z_i}}{c \cdot \rho \cdot \dot{V}_z} \tag{7}$$
+$$T_{z_1} = \frac{\dot{H}_{z_i}}{c \cdot \rho \cdot \dot{V}_z} \tag{52}$$
 with
-$$\dot{H}_{z_i} = \Delta \dot{H}_{z_{0,1}} + \dot{H}_{z_0} = \rho \cdot c \cdot \dot{V}_z \cdot (T_{z1,channel} - T_{z,0}) + \rho \cdot c \cdot \dot{V}_x \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{thread}) + \rho \cdot c \cdot \dot{V}_{groove} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{8}$$
+$$\dot{H}_{z_i} = \Delta \dot{H}_{z_{0,1}} + \dot{H}_{z_0} = \rho \cdot c \cdot \dot{V}_z \cdot (T_{z1,channel} - T_{z,0}) + \rho \cdot c \cdot \dot{V}_x \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{thread}) + \rho \cdot c \cdot \dot{V}_{groove} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{53}$$
 one eventually receives the following equation for the calculation of the temperature development:
-$$T_{z_1} = T_{z1,channel} + \frac{\dot{V}_x}{\dot{V}_z} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) + \frac{\dot{V}_{groove}}{\dot{V}_z} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{9}$$
+$$T_{z_1} = T_{z1,channel} + \frac{\dot{V}_x}{\dot{V}_z} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) + \frac{\dot{V}_{groove}}{\dot{V}_z} \cdot (\overline{T_{z01,i+1}} - \overline{T_{z01,i}} + \Delta T_{groove}) \tag{54}$$
 First experiments for the verification of the temperature model were performed with a twin screw extruder of the ZSK 30 type. A polypropylene (PP 1100H) served as the experimental medium. The polymer was plasticated in a subsidiary extruder and laterally conveyed into the twin screw extruder (see figure).
@@ Zeile 310: / Zeile 310: @@
 It can be recognized that the model is in a position to describe the experimental results with sufficient accuracy.
+{{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_massetemperatur_007.svg?nolink&700 |}}
 **Figure:** Comparison of measured and calculated melt temperatures in thread elements.
@@ Zeile 317: / Zeile 319: @@
 The energy equation for the calculation of the radial temperature profile in the screw flight forms the basis for the 2D model.
-$$\rho c \left(\frac{\partial T}{\partial t} + v \nabla T\right) = \lambda \nabla^2 T + \tau \nabla v \tag{Equation 1}$$
+$$\rho c \left(\frac{\partial T}{\partial t} + v \nabla T\right) = \lambda \nabla^2 T + \tau \nabla v \tag{55}$$
 A modification of the energy equation to match the existing open system, which exchanges the energy and the mass with the environment, follows.
@@ Zeile 327: / Zeile 329: @@
 $$-T \left(\frac{\partial p}{\partial T}\right)_V \left(\frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}\right) - \left(\sigma_{xx} \frac{\partial v_x}{\partial x} + \sigma_{yy} \frac{\partial v_y}{\partial y} + \sigma_{zz} \frac{\partial v_z}{\partial z}\right) $$
-$$-\left[\tau_{xy} \left(\frac{\partial v_y}{\partial x} + \frac{\partial v_x}{\partial y}\right) + \tau_{xz} \left(\frac{\partial v_z}{\partial x} + \frac{\partial v_x}{\partial z}\right) + \tau_{yz} \left(\frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y}\right)\right]\tag{Equation 2}$$
+$$-\left[\tau_{xy} \left(\frac{\partial v_y}{\partial x} + \frac{\partial v_x}{\partial y}\right) + \tau_{xz} \left(\frac{\partial v_z}{\partial x} + \frac{\partial v_x}{\partial z}\right) + \tau_{yz} \left(\frac{\partial v_y}{\partial z} + \frac{\partial v_z}{\partial y}\right)\right]\tag{56}$$
 The equation consists of five terms, which have the following meanings [[en:grundlagenhandbuch:massetemperatur#references |[Ang10]]]:
@@ Zeile 348: / Zeile 350: @@
   * Normal stresses are also neglectable ($\sigma_{xx} = \sigma_{yy} = \sigma_{zz} = 0$)
-With the help of the simplifying criteria the equation 4-2 can be reduced to 3 terms:
+With the help of the simplifying criteria the equation 56 can be reduced:
-$$\rho c v_z \frac{\partial T}{\partial z} = -\frac{\partial \dot{q}_y}{\partial y} - \tau_{yz} \left(\frac{\partial v_z}{\partial y}\right)\tag{Equation 3}$$
+$$\rho c v_z \frac{\partial T}{\partial z} = -\frac{\partial \dot{q}_y}{\partial y} - \tau_{yz} \left(\frac{\partial v_z}{\partial y}\right)\tag{57}$$
 The sum of thermal conduction in the flight height and the energy of the dissipation results in the temperature rising lengthways in the canal, which is filled with polymer melt.
-Under consideration of the Fourier thermal conductivity approach for the heat flow in direction of the flight height and the power flow law for non-newtonian flow behavior of polymer melt with the Arrehnius-approach, the equation 4-3 can be extended to the following:
+Under consideration of the Fourier thermal conductivity approach for the heat flow in direction of the flight height and the power flow law for non-newtonian flow behavior of polymer melt with the Arrehnius-approach, the equation 57 can be extended to the following:
-$$\rho c \bar{v}_z \frac{\partial T}{\partial z} = \lambda \frac{\partial^2 T}{\partial y^2} + \left(\overline{\tau\dot{\gamma}}\right)_0 e^{-\beta(T-T_0)}\tag{Equation 4}$$
+$$\rho c \bar{v}_z \frac{\partial T}{\partial z} = \lambda \frac{\partial^2 T}{\partial y^2} + \left(\overline{\tau\dot{\gamma}}\right)_0 e^{-\beta(T-T_0)}\tag{58}$$
 To simplify the energy equation further, the dimensionless operating figures as described in [[en:grundlagenhandbuch:massetemperatur#references |[Ang10]]], the Graetz- ($Gz$) and the Brinkmann-number ($Br$), are used. Herein, the Brinkmann-number describes the relation of disperse energy in the screw canal to the heat conduction in the direction of the flight height. The Graetz-number describes the convection in the direction of the flight length to the heat conduction in the direction of the flight height. Additionally, the dimensionless coordinates $\xi$, $\zeta$ and the dimensionless temperature $\Theta$ are placed in the energy equation.
-$$Br = \frac{\left(\overline{\tau\dot{\gamma}}\right)_0 h^2}{\lambda T_Z} \approx \frac{K_{0T} v_0^{1+n} h^{1-n}}{\lambda T_Z}\tag{Equation 5}$$
+$$Br = \frac{\left(\overline{\tau\dot{\gamma}}\right)_0 h^2}{\lambda T_Z} \approx \frac{K_{0T} v_0^{1+n} h^{1-n}}{\lambda T_Z}\tag{59}$$
-$$Gz = \frac{c \rho \bar{v}_z h^2}{\lambda \Delta z} = \frac{c \rho h}{\lambda b \Delta z} \dot{V}\tag{Equation 6}$$
+$$Gz = \frac{c \rho \bar{v}_z h^2}{\lambda \Delta z} = \frac{c \rho h}{\lambda b \Delta z} \dot{V}\tag{60}$$
-$$\Theta = \frac{T - T_0}{T_Z}\tag{Equation 7}$$
+$$\Theta = \frac{T - T_0}{T_Z}\tag{61}$$
-$$\xi = \frac{\Delta y}{h}\tag{Equation 8}$$
+$$\xi = \frac{\Delta y}{h}\tag{62}$$
-$$\zeta = \frac{\Delta z}{l}\tag{Equation 9}$$
+$$\zeta = \frac{\Delta z}{l}\tag{63}$$
 During the geometric calculation of the Graetz- $Gr$ and Brinkmann-number $Br$ some twin screw-specific adjustments are made. For the ascertainment of the Graetz-number $Gr$ the flight height $h$ and the flight width $br$ are important. These two sizes are defined using the average flight height $\bar{h}$ and the flight width $b_{max}$ of the twin screw extruder.
@@ Zeile 376: / Zeile 378: @@
 For the Brinkmann-number $Br$ the viscosity $\eta$ is needed, this depends on the shear rate $\dot{\gamma}$ and is influenced by the geometry of the twin screw extruder. As there is no constant flight height $h$ in the canal, the average flight height $\bar{h}$ is used to calculate the average shear rate $\overline{\dot{\gamma}}$. Without considering the influence of the characteristics of the screw element (mixing elements, Shear elements etc.) [[en:grundlagenhandbuch:massetemperatur#references |[Kre04]]].
-$$\overline{\dot{\gamma}} = \frac{\bar{h}}{v_0}\tag{Equation 10}$$
+$$\overline{\dot{\gamma}} = \frac{\bar{h}}{v_0}\tag{64}$$
 The function for the calculation of the temperature in the melt-filled canal, assumes the following shape:
-$$\frac{\partial \Theta}{\partial \xi} = \frac{1}{Gz} \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} \exp\left[-\beta(T_Z \Theta)\right]\tag{Equation 11}$$
+$$\frac{\partial \Theta}{\partial \xi} = \frac{1}{Gz} \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{Br}{Gz} \exp\left[-\beta(T_Z \Theta)\right]\tag{65}$$
 It is not possible yet, to solve the, as the exponential term is dependent on the temperature $T$ or rather $\Theta$. In the following equation the exponential term is partly linearized.
-$$\exp\left[-\beta(T - T_0)\right] = \exp\left[-\beta(T_Z \Theta)\right] = c_1 - c_2 \beta T_Z \Theta\tag{Equation 12}$$
+$$\exp\left[-\beta(T - T_0)\right] = \exp\left[-\beta(T_Z \Theta)\right] = c_1 - c_2 \beta T_Z \Theta\tag{66}$$
 This linear equation divides the exponential term into two parts. The first part is dependent on the radial temperature $T$ or rather $\Theta$ ($c_2 \beta T_Z \Theta$) and the second is dependent on $T$ or rather $\Theta$ and $c_1$. The two unknown parameters $c_1$ and $c_2$ are calculated in two equations, which approach the exponential function with a secant which lies between the average radial temperature of the previous section $T_0$ and the cylinder wall temperature $T_Z$.
@@ Zeile 394: / Zeile 396: @@
 In summary, the linearization with the Equation 10 produces the analytically solvable, simplified energy equation for the radial temperature (for $0 \leq \xi \leq 1$ and $0 \leq \zeta \leq 1$) [[en:grundlagenhandbuch:massetemperatur#references |[Ang10]]]:
-$$\frac{\partial^2 \Theta}{\partial \xi^2} - Gz \frac{\partial \Theta}{\partial \xi} - c_2 \beta T_Z Br \Theta = -c_1 Br\tag{Equation 13}$$
+$$\frac{\partial^2 \Theta}{\partial \xi^2} - Gz \frac{\partial \Theta}{\partial \xi} - c_2 \beta T_Z Br \Theta = -c_1 Br\tag{67}$$
 The presented differential equation was successfully solved with the marginal conditions of the constant cylinder temperature and a tempered screw in the elaboration of [[en:grundlagenhandbuch:massetemperatur#references |[Sch13]]]. This model was adjusted according to the calculation of the temperature progression in the co-rotating twin screw extruder. As the solution contains dimensionless temperature data, these must still be converted into dimensioned data.
@@ Zeile 400: / Zeile 402: @@
 This can be done using the following equation and results in the ten data of the profile of the flight height.
-$$T_{0,n}(\xi) = \theta_{\xi,n} T_{Z,n} + T_{n-1}\tag{Equation 14}$$
+$$T_{0,n}(\xi) = \theta_{\xi,n} T_{Z,n} + T_{n-1}\tag{68}$$
 Finally the arithmetic mean is formed from the radial temperature data and the average canal temperature of the section is defined. Because of the step-by-step calculation in SIGMA the axial temperature curve can be calculated.